Red Flag was born from heavy Thud losses, primarily, and the mission concept has always been to provide a realistic combat environment to train in so the inevitable mistakes will not teach their lessons in reality.
USAF training doctrine always held (until the late 90s) that a UPT grad was universally assignable, hence tanker pilots and such were thrown into the Thud and sent into combat after a short RTU exposure. The 'fighter-qualified' label only mattered for the initial assignment out of UPT.
The process was reversed in my case, as I was involuntarily shunted from RF-4s to the KC-135. Bitter at first, I quickly realized it was the best thing that could have happened to me in the air force. I stayed on the tanker in the reserves for the remaining 26 of my 30 years.